# System equations and abstractions

[From Bill Powers (931214.1950 MST(]

Bob Clark, Martin Taylor (931214) --

Both of you questioned my statement that energy can be "reduced
to" force and distance, but not vice versa. Perhaps I used the
wrong term. I meant that energy can be expressed as a function of
force and distance, but neither force nor distance can be
expressed as a function of energy.

In any differential equation in which terms appear that have a
product of something like a force (say, the length of a spring),
integrated over a distance, those terms can be picked out and
identified as "energy terms." Some of those terms might involve
one sign of energy, as when a spring is being compressed: the
integral is called potential energy. Other terms might involve
for example deceleration of moving objects: conversion of the the
opposite sign of energy, kinetic energy, into some other form.
When a moving object is stopped by a spring, we can refer to
appropriate parts of the equations describing this event as
"kinetic energy being converted into potential energy." We can
say that the energy lost by one part of the system is gained by
another part.

But what is it that makes the physical system work as it does?
All that is required to explain that is the set of differential
equations showing how forces act on masses and deformable bodies.
That description is complete; that is, nothing is added to it by
picking out the energy terms and speaking metaphorically to the
effect that something called energy has been transmitted from one
form into another. That mode of description is an abstraction: it
picks out certain combinations of terms to represent as a single
variable. If all the terms in an equation are abstracted in this
way, we have a description with fewer degrees of freedom, which
does not contain enough information to reconstruct the original
situation from which the abstraction was obtained. This is
inevitable: any time you choose to represent a sum or product as
a single variable, you have lost the values of the individual
variables, and can't get back.

There are no _separate_ energy equations. The energy equations of
a behaving system are simply the equations of the behaving
system. We call certain groups of elements of those equations
"energy." But if we don't do that, we still have the description
of the same system, with nothing lost (except, perhaps, certain
convenient modes of computation). To create a description in
terms of energy, we do not have to introduce any new variables;
all that we need is already there in the behavioral equations.

Everything I have said about energy is equally true of all other
abstract ways of describing a physical system. Abstractions never